Statics and dynamics of coupled electron-phonon systems

We review recent variational methods that provide extremely accurate soluti ons for the problem of one and two electrons coupled to dynamical, quantum phonons, including retardation effects. The dynamics of these systems far f rom equilibrium is explored, including the dynamics of polaron formation af ter electron injection by an STM. An examination of three-point functions s uggests a qualitative modification of the traditional picture of a polaron. Long-range electron-phonon interactions are studied in some generality, in cluding vector and frustrated Frohlich models, and bounds are obtained on t he possible mass enhancement.